TESTING/EIG/schkst2stg.f(3)

NAME
TESTING/EIG/schkst2stg.f

SYNOPSIS

  Functions/Subroutines
subroutine schkst2stg (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, ap, sd, se, d1, d2, d3, d4, d5, wa1, wa2, wa3, wr, u, ldu, v, vp, tau, z, work, lwork, iwork, liwork, result, info)
SCHKST2STG

Function/Subroutine Documentation

  subroutine schkst2stg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( * ) ap, real, dimension( * ) sd, real, dimension( * ) se, real, dimension( * ) d1, real, dimension( * ) d2, real, dimension( * ) d3, real, dimension( * ) d4, real, dimension( * ) d5, real, dimension( * ) wa1, real, dimension( * ) wa2, real, dimension( * ) wa3, real, dimension( * ) wr, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) v, real, dimension( * ) vp, real, dimension( * ) tau, real, dimension( ldu, * ) z, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, real, dimension( * ) result, integer info)

SCHKST2STG

Purpose:

    !>
    !> SCHKST2STG  checks the symmetric eigenvalue problem routines
    !> using the 2-stage reduction techniques. Since the generation
    !> of Q or the vectors is not available in this release, we only
    !> compare the eigenvalue resulting when using the 2-stage to the
    !> one considered as reference using the standard 1-stage reduction
    !> SSYTRD. For that, we call the standard SSYTRD and compute D1 using
    !> SSTEQR, then we call the 2-stage SSYTRD_2STAGE with Upper and Lower
    !> and we compute D2 and D3 using SSTEQR and then we replaced tests
    !> 3 and 4 by tests 11 and 12. test 1 and 2 remain to verify that
    !> the 1-stage results are OK and can be trusted.
    !> This testing routine will converge to the SCHKST in the next
    !> release when vectors and generation of Q will be implemented.
    !>
    !>    SSYTRD factors A as  U S U' , where ' means transpose,
    !>    S is symmetric tridiagonal, and U is orthogonal.
    !>    SSYTRD can use either just the lower or just the upper triangle
    !>    of A; SCHKST2STG checks both cases.
    !>    U is represented as a product of Householder
    !>    transformations, whose vectors are stored in the first
    !>    n-1 columns of V, and whose scale factors are in TAU.
    !>
    !>    SSPTRD does the same as SSYTRD, except that A and V are stored
    !>    in  format.
    !>
    !>    SORGTR constructs the matrix U from the contents of V and TAU.
    !>
    !>    SOPGTR constructs the matrix U from the contents of VP and TAU.
    !>
    !>    SSTEQR factors S as  Z D1 Z' , where Z is the orthogonal
    !>    matrix of eigenvectors and D1 is a diagonal matrix with
    !>    the eigenvalues on the diagonal.  D2 is the matrix of
    !>    eigenvalues computed when Z is not computed.
    !>
    !>    SSTERF computes D3, the matrix of eigenvalues, by the
    !>    PWK method, which does not yield eigenvectors.
    !>
    !>    SPTEQR factors S as  Z4 D4 Z4' , for a
    !>    symmetric positive definite tridiagonal matrix.
    !>    D5 is the matrix of eigenvalues computed when Z is not
    !>    computed.
    !>
    !>    SSTEBZ computes selected eigenvalues.  WA1, WA2, and
    !>    WA3 will denote eigenvalues computed to high
    !>    absolute accuracy, with different range options.
    !>    WR will denote eigenvalues computed to high relative
    !>    accuracy.
    !>
    !>    SSTEIN computes Y, the eigenvectors of S, given the
    !>    eigenvalues.
    !>
    !>    SSTEDC factors S as Z D1 Z' , where Z is the orthogonal
    !>    matrix of eigenvectors and D1 is a diagonal matrix with
    !>    the eigenvalues on the diagonal ('I' option). It may also
    !>    update an input orthogonal matrix, usually the output
    !>    from SSYTRD/SORGTR or SSPTRD/SOPGTR ('V' option). It may
    !>    also just compute eigenvalues ('N' option).
    !>
    !>    SSTEMR factors S as Z D1 Z' , where Z is the orthogonal
    !>    matrix of eigenvectors and D1 is a diagonal matrix with
    !>    the eigenvalues on the diagonal ('I' option).  SSTEMR
    !>    uses the Relatively Robust Representation whenever possible.
    !>
    !> When SCHKST2STG is called, a number of matrix  () and a
    !> number of matrix  are specified.  For each size ()
    !> and each type of matrix, one matrix will be generated and used
    !> to test the symmetric eigenroutines.  For each matrix, a number
    !> of tests will be performed:
    !>
    !> (1)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='U', ... )
    !>
    !> (2)     | I - UV' | / ( n ulp )        SORGTR( UPLO='U', ... )
    !>
    !> (3)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='L', ... )
    !>         replaced by | D1 - D2 | / ( |D1| ulp ) where D1 is the
    !>         eigenvalue matrix computed using S and D2 is the
    !>         eigenvalue matrix computed using S_2stage the output of
    !>         SSYTRD_2STAGE(, ,....). D1 and D2 are computed
    !>         via SSTEQR('N',...)
    !>
    !> (4)     | I - UV' | / ( n ulp )        SORGTR( UPLO='L', ... )
    !>         replaced by | D1 - D3 | / ( |D1| ulp ) where D1 is the
    !>         eigenvalue matrix computed using S and D3 is the
    !>         eigenvalue matrix computed using S_2stage the output of
    !>         SSYTRD_2STAGE(, ,....). D1 and D3 are computed
    !>         via SSTEQR('N',...)
    !>
    !> (5-8)   Same as 1-4, but for SSPTRD and SOPGTR.
    !>
    !> (9)     | S - Z D Z' | / ( |S| n ulp ) SSTEQR('V',...)
    !>
    !> (10)    | I - ZZ' | / ( n ulp )        SSTEQR('V',...)
    !>
    !> (11)    | D1 - D2 | / ( |D1| ulp )        SSTEQR('N',...)
    !>
    !> (12)    | D1 - D3 | / ( |D1| ulp )        SSTERF
    !>
    !> (13)    0 if the true eigenvalues (computed by sturm count)
    !>         of S are within THRESH of
    !>         those in D1.  2*THRESH if they are not.  (Tested using
    !>         SSTECH)
    !>
    !> For S positive definite,
    !>
    !> (14)    | S - Z4 D4 Z4' | / ( |S| n ulp ) SPTEQR('V',...)
    !>
    !> (15)    | I - Z4 Z4' | / ( n ulp )        SPTEQR('V',...)
    !>
    !> (16)    | D4 - D5 | / ( 100 |D4| ulp )       SPTEQR('N',...)
    !>
    !> When S is also diagonally dominant by the factor gamma < 1,
    !>
    !> (17)    max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
    !>          i
    !>         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
    !>                                              SSTEBZ( 'A', 'E', ...)
    !>
    !> (18)    | WA1 - D3 | / ( |D3| ulp )          SSTEBZ( 'A', 'E', ...)
    !>
    !> (19)    ( max { min | WA2(i)-WA3(j) | } +
    !>            i     j
    !>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
    !>            i     j
    !>                                              SSTEBZ( 'I', 'E', ...)
    !>
    !> (20)    | S - Y WA1 Y' | / ( |S| n ulp )  SSTEBZ, SSTEIN
    !>
    !> (21)    | I - Y Y' | / ( n ulp )          SSTEBZ, SSTEIN
    !>
    !> (22)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('I')
    !>
    !> (23)    | I - ZZ' | / ( n ulp )           SSTEDC('I')
    !>
    !> (24)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('V')
    !>
    !> (25)    | I - ZZ' | / ( n ulp )           SSTEDC('V')
    !>
    !> (26)    | D1 - D2 | / ( |D1| ulp )           SSTEDC('V') and
    !>                                              SSTEDC('N')
    !>
    !> Test 27 is disabled at the moment because SSTEMR does not
    !> guarantee high relatvie accuracy.
    !>
    !> (27)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
    !>          i
    !>         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
    !>                                              SSTEMR('V', 'A')
    !>
    !> (28)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
    !>          i
    !>         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
    !>                                              SSTEMR('V', 'I')
    !>
    !> Tests 29 through 34 are disable at present because SSTEMR
    !> does not handle partial spectrum requests.
    !>
    !> (29)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'I')
    !>
    !> (30)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'I')
    !>
    !> (31)    ( max { min | WA2(i)-WA3(j) | } +
    !>            i     j
    !>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
    !>            i     j
    !>         SSTEMR('N', 'I') vs. SSTEMR('V', 'I')
    !>
    !> (32)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'V')
    !>
    !> (33)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'V')
    !>
    !> (34)    ( max { min | WA2(i)-WA3(j) | } +
    !>            i     j
    !>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
    !>            i     j
    !>         SSTEMR('N', 'V') vs. SSTEMR('V', 'V')
    !>
    !> (35)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'A')
    !>
    !> (36)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'A')
    !>
    !> (37)    ( max { min | WA2(i)-WA3(j) | } +
    !>            i     j
    !>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
    !>            i     j
    !>         SSTEMR('N', 'A') vs. SSTEMR('V', 'A')
    !>
    !> The  are specified by an array NN(1:NSIZES); the value of
    !> each element NN(j) specifies one size.
    !> The  are specified by a logical array DOTYPE( 1:NTYPES );
    !> if DOTYPE(j) is .TRUE., then matrix type  will be generated.
    !> Currently, the list of possible types is:
    !>
    !> (1)  The zero matrix.
    !> (2)  The identity matrix.
    !>
    !> (3)  A diagonal matrix with evenly spaced entries
    !>      1, ..., ULP  and random signs.
    !>      (ULP = (first number larger than 1) - 1 )
    !> (4)  A diagonal matrix with geometrically spaced entries
    !>      1, ..., ULP  and random signs.
    !> (5)  A diagonal matrix with  entries 1, ULP, ..., ULP
    !>      and random signs.
    !>
    !> (6)  Same as (4), but multiplied by SQRT( overflow threshold )
    !> (7)  Same as (4), but multiplied by SQRT( underflow threshold )
    !>
    !> (8)  A matrix of the form  U' D U, where U is orthogonal and
    !>      D has evenly spaced entries 1, ..., ULP with random signs
    !>      on the diagonal.
    !>
    !> (9)  A matrix of the form  U' D U, where U is orthogonal and
    !>      D has geometrically spaced entries 1, ..., ULP with random
    !>      signs on the diagonal.
    !>
    !> (10) A matrix of the form  U' D U, where U is orthogonal and
    !>      D has  entries 1, ULP,..., ULP with random
    !>      signs on the diagonal.
    !>
    !> (11) Same as (8), but multiplied by SQRT( overflow threshold )
    !> (12) Same as (8), but multiplied by SQRT( underflow threshold )
    !>
    !> (13) Symmetric matrix with random entries chosen from (-1,1).
    !> (14) Same as (13), but multiplied by SQRT( overflow threshold )
    !> (15) Same as (13), but multiplied by SQRT( underflow threshold )
    !> (16) Same as (8), but diagonal elements are all positive.
    !> (17) Same as (9), but diagonal elements are all positive.
    !> (18) Same as (10), but diagonal elements are all positive.
    !> (19) Same as (16), but multiplied by SQRT( overflow threshold )
    !> (20) Same as (16), but multiplied by SQRT( underflow threshold )
    !> (21) A diagonally dominant tridiagonal matrix with geometrically
    !>      spaced diagonal entries 1, ..., ULP.
    !>

Parameters
    NSIZES

    !>          NSIZES is INTEGER
    !>          The number of sizes of matrices to use.  If it is zero,
    !>          SCHKST2STG does nothing.  It must be at least zero.
    !>

    NN

    !>          NN is INTEGER array, dimension (NSIZES)
    !>          An array containing the sizes to be used for the matrices.
    !>          Zero values will be skipped.  The values must be at least
    !>          zero.
    !>

    NTYPES

    !>          NTYPES is INTEGER
    !>          The number of elements in DOTYPE.   If it is zero, SCHKST2STG
    !>          does nothing.  It must be at least zero.  If it is MAXTYP+1
    !>          and NSIZES is 1, then an additional type, MAXTYP+1 is
    !>          defined, which is to use whatever matrix is in A.  This
    !>          is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
    !>          DOTYPE(MAXTYP+1) is .TRUE. .
    !>

    DOTYPE

    !>          DOTYPE is LOGICAL array, dimension (NTYPES)
    !>          If DOTYPE(j) is .TRUE., then for each size in NN a
    !>          matrix of that size and of type j will be generated.
    !>          If NTYPES is smaller than the maximum number of types
    !>          defined (PARAMETER MAXTYP), then types NTYPES+1 through
    !>          MAXTYP will not be generated.  If NTYPES is larger
    !>          than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
    !>          will be ignored.
    !>

    ISEED

    !>          ISEED is INTEGER array, dimension (4)
    !>          On entry ISEED specifies the seed of the random number
    !>          generator. The array elements should be between 0 and 4095;
    !>          if not they will be reduced mod 4096.  Also, ISEED(4) must
    !>          be odd.  The random number generator uses a linear
    !>          congruential sequence limited to small integers, and so
    !>          should produce machine independent random numbers. The
    !>          values of ISEED are changed on exit, and can be used in the
    !>          next call to SCHKST2STG to continue the same random number
    !>          sequence.
    !>

    THRESH

    !>          THRESH is REAL
    !>          A test will count as  if the , computed as
    !>          described above, exceeds THRESH.  Note that the error
    !>          is scaled to be O(1), so THRESH should be a reasonably
    !>          small multiple of 1, e.g., 10 or 100.  In particular,
    !>          it should not depend on the precision (single vs. double)
    !>          or the size of the matrix.  It must be at least zero.
    !>

    NOUNIT

    !>          NOUNIT is INTEGER
    !>          The FORTRAN unit number for printing out error messages
    !>          (e.g., if a routine returns IINFO not equal to 0.)
    !>

    A

    !>          A is REAL array of
    !>                                  dimension ( LDA , max(NN) )
    !>          Used to hold the matrix whose eigenvalues are to be
    !>          computed.  On exit, A contains the last matrix actually
    !>          used.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of A.  It must be at
    !>          least 1 and at least max( NN ).
    !>

    AP

    !>          AP is REAL array of
    !>                      dimension( max(NN)*max(NN+1)/2 )
    !>          The matrix A stored in packed format.
    !>

    SD

    !>          SD is REAL array of
    !>                             dimension( max(NN) )
    !>          The diagonal of the tridiagonal matrix computed by SSYTRD.
    !>          On exit, SD and SE contain the tridiagonal form of the
    !>          matrix in A.
    !>

    SE

    !>          SE is REAL array of
    !>                             dimension( max(NN) )
    !>          The off-diagonal of the tridiagonal matrix computed by
    !>          SSYTRD.  On exit, SD and SE contain the tridiagonal form of
    !>          the matrix in A.
    !>

    D1

    !>          D1 is REAL array of
    !>                             dimension( max(NN) )
    !>          The eigenvalues of A, as computed by SSTEQR simultaneously
    !>          with Z.  On exit, the eigenvalues in D1 correspond with the
    !>          matrix in A.
    !>

    D2

    !>          D2 is REAL array of
    !>                             dimension( max(NN) )
    !>          The eigenvalues of A, as computed by SSTEQR if Z is not
    !>          computed.  On exit, the eigenvalues in D2 correspond with
    !>          the matrix in A.
    !>

    D3

    !>          D3 is REAL array of
    !>                             dimension( max(NN) )
    !>          The eigenvalues of A, as computed by SSTERF.  On exit, the
    !>          eigenvalues in D3 correspond with the matrix in A.
    !>

    D4

    !>          D4 is REAL array of
    !>                             dimension( max(NN) )
    !>          The eigenvalues of A, as computed by SPTEQR(V).
    !>          SPTEQR factors S as  Z4 D4 Z4*
    !>          On exit, the eigenvalues in D4 correspond with the matrix in A.
    !>

    D5

    !>          D5 is REAL array of
    !>                             dimension( max(NN) )
    !>          The eigenvalues of A, as computed by SPTEQR(N)
    !>          when Z is not computed. On exit, the
    !>          eigenvalues in D4 correspond with the matrix in A.
    !>

    WA1

    !>          WA1 is REAL array of
    !>                             dimension( max(NN) )
    !>          All eigenvalues of A, computed to high
    !>          absolute accuracy, with different range options.
    !>          as computed by SSTEBZ.
    !>

    WA2

    !>          WA2 is REAL array of
    !>                             dimension( max(NN) )
    !>          Selected eigenvalues of A, computed to high
    !>          absolute accuracy, with different range options.
    !>          as computed by SSTEBZ.
    !>          Choose random values for IL and IU, and ask for the
    !>          IL-th through IU-th eigenvalues.
    !>

    WA3

    !>          WA3 is REAL array of
    !>                             dimension( max(NN) )
    !>          Selected eigenvalues of A, computed to high
    !>          absolute accuracy, with different range options.
    !>          as computed by SSTEBZ.
    !>          Determine the values VL and VU of the IL-th and IU-th
    !>          eigenvalues and ask for all eigenvalues in this range.
    !>

    WR

    !>          WR is REAL array of
    !>                             dimension( max(NN) )
    !>          All eigenvalues of A, computed to high
    !>          absolute accuracy, with different options.
    !>          as computed by SSTEBZ.
    !>

    U

    !>          U is REAL array of
    !>                             dimension( LDU, max(NN) ).
    !>          The orthogonal matrix computed by SSYTRD + SORGTR.
    !>

    LDU

    !>          LDU is INTEGER
    !>          The leading dimension of U, Z, and V.  It must be at least 1
    !>          and at least max( NN ).
    !>

    V

    !>          V is REAL array of
    !>                             dimension( LDU, max(NN) ).
    !>          The Housholder vectors computed by SSYTRD in reducing A to
    !>          tridiagonal form.  The vectors computed with UPLO='U' are
    !>          in the upper triangle, and the vectors computed with UPLO='L'
    !>          are in the lower triangle.  (As described in SSYTRD, the
    !>          sub- and superdiagonal are not set to 1, although the
    !>          true Householder vector has a 1 in that position.  The
    !>          routines that use V, such as SORGTR, set those entries to
    !>          1 before using them, and then restore them later.)
    !>

    VP

    !>          VP is REAL array of
    !>                      dimension( max(NN)*max(NN+1)/2 )
    !>          The matrix V stored in packed format.
    !>

    TAU

    !>          TAU is REAL array of
    !>                             dimension( max(NN) )
    !>          The Householder factors computed by SSYTRD in reducing A
    !>          to tridiagonal form.
    !>

    Z

    !>          Z is REAL array of
    !>                             dimension( LDU, max(NN) ).
    !>          The orthogonal matrix of eigenvectors computed by SSTEQR,
    !>          SPTEQR, and SSTEIN.
    !>

    WORK

    !>          WORK is REAL array of
    !>                      dimension( LWORK )
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The number of entries in WORK.  This must be at least
    !>          1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2
    !>          where Nmax = max( NN(j), 2 ) and lg = log base 2.
    !>

    IWORK

    !>          IWORK is INTEGER array,
    !>          Workspace.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The number of entries in IWORK.  This must be at least
    !>                  6 + 6*Nmax + 5 * Nmax * lg Nmax
    !>          where Nmax = max( NN(j), 2 ) and lg = log base 2.
    !>

    RESULT

    !>          RESULT is REAL array, dimension (26)
    !>          The values computed by the tests described above.
    !>          The values are currently limited to 1/ulp, to avoid
    !>          overflow.
    !>

    INFO

    !>          INFO is INTEGER
    !>          If 0, then everything ran OK.
    !>           -1: NSIZES < 0
    !>           -2: Some NN(j) < 0
    !>           -3: NTYPES < 0
    !>           -5: THRESH < 0
    !>           -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
    !>          -23: LDU < 1 or LDU < NMAX.
    !>          -29: LWORK too small.
    !>          If  SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF,
    !>              or SORMC2 returns an error code, the
    !>              absolute value of it is returned.
    !>
    !>-----------------------------------------------------------------------
    !>
    !>       Some Local Variables and Parameters:
    !>       ---- ----- --------- --- ----------
    !>       ZERO, ONE       Real 0 and 1.
    !>       MAXTYP          The number of types defined.
    !>       NTEST           The number of tests performed, or which can
    !>                       be performed so far, for the current matrix.
    !>       NTESTT          The total number of tests performed so far.
    !>       NBLOCK          Blocksize as returned by ENVIR.
    !>       NMAX            Largest value in NN.
    !>       NMATS           The number of matrices generated so far.
    !>       NERRS           The number of tests which have exceeded THRESH
    !>                       so far.
    !>       COND, IMODE     Values to be passed to the matrix generators.
    !>       ANORM           Norm of A; passed to matrix generators.
    !>
    !>       OVFL, UNFL      Overflow and underflow thresholds.
    !>       ULP, ULPINV     Finest relative precision and its inverse.
    !>       RTOVFL, RTUNFL  Square roots of the previous 2 values.
    !>               The following four arrays decode JTYPE:
    !>       KTYPE(j)        The general type (1-10) for type .
    !>       KMODE(j)        The MODE value to be passed to the matrix
    !>                       generator for type .
    !>       KMAGN(j)        The order of magnitude ( O(1),
    !>                       O(overflow^(1/2) ), O(underflow^(1/2) )
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 608 of file schkst2stg.f.

Author

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